As you are likely aware Dear Reader, due to the accident and intent of where I tend to sit on the scientific spectrum, the scooping type of competition is not a huge part of my professional life. That is, I have managed to get by to this point by not being terribly afraid of people knowing what I am working on or what I plan to work on. Part of this has to do with playing at a level of publication that is not obsessed with the very first person to demonstrate something. Part of it is selecting research questions that are not densely populated with dozens or scores of other laboratories trying to scratch the same flea. Part of it is my overweening and misplaced self-confidence that we did it better, dammit, so who cares who published first.

Part of it is pure wrongheadedness on my part, no doubt.

When it comes to grants, specifically, I was always around people who were reflexively generous with sharing their applications when I was a late-postdoc and an early-career faculty member. As time has gone on and more people are asking me for my proposals than I feel the need to ask, I have given mine out to anyone who requests them. (Usually with a little lecture about how my "successful" apps are no more informative than my triaged ones, of course.)

So take that into account.

On a purely tactical level, it is possible for the postdoc in this situation to simply refuse. We can extend this to PIs who are asked for their successful grant applications. You can just say no.

It seems to me to be unwise to do so, particularly when it comes to an application that has been successful. Even if you cannot stand the person who is asking. It just seem churlish when the cost to you is so low.

Is it going to give this person ScienceEnemy little boost ahead? Sure. But remember, the odds of funding are still very steep. So it isn't like you are handing them an award. They still have to write a credible application. And get lucky. So why not*? It costs you essentially nothing to email over your application.

On a strategic level, this person could be your colleague in science for a long time. They could very well be in a position to review you and your work, particularly if they are in a related area of science. And even if they annoy you, it isn't necessarily the case that they have so much as noticed. Lots of annoying people are kind of unaware... So why make an enemy?

And there is one more thing to consider. If you act within a professional capacity on personal whim and dislike, what does this say about your behavior as an objective peer reviewer? Shouldn't you be able to set aside personal dislike to effectively review the scientific content of a paper or grant proposal? Yes, yes you should.

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*Now, if you think the person is a data fraud or something...well that is entirely different.

DM, there are a few more considerations for sharing:
i) Grants involve a lot of thinking and are a concise presentation of the data and our best arguments for why something is important. Handing it over to another lab where they could use their data to scoop you with your own hypothesis is not a good idea.
ii) Often it is not just the data or the argument, but the way these were presented. In my lab we spend a lot of time "mining" our data and looking for the clearest possible way to support a point or test a hypothesis. I don't mind a competitor copying the approach... as long as they cite our paper.
iii) I would only share with people that share. I don't share with those labs that refused to share or pulled tricks when I asked for something (or when someone else did).

The older the grant, the more likely I don't mind sharing.

Still, the most compelling reason to share is the one you put forward: what goes around, comes around.

Winning grant applications are public documents, HOWEVER, there are rules about what must be shared and what not. There is an NIH policy which you would be perfectly within your rights to follow.

Since the successful grant application can be obtained by FOIA from NIH or other agencies, you might as well share it with redactions. Doing so gives you a bit more freedom, for example you could only provide the abstract, text and references. For example, you could provide a print copy rather than an Adobe Acrobat copy, etc.

Unfunded applications are NOT public documents. Summary statements are, without scores, but Eli would direct anyone who asked to NIH for those.

One other consideration: some people will , without malintent, liberally forward successful grants that have been given to them to third parties. I would always tell them not to share my application with somebody else. Also because of this, I may give sometimes only a printout.

Bunbury-
I totally agree. It is rude to share someone else's work without their explicit permission. Each time, unless they have given you blanket permission to do so.

It gets a little trickier when you have, for example, shared applications within a lab for so long that it is not clear which of the prior three or four generations of postdocs (or PIs) contributed which particular bit of stylistic brilliance.

I am happy to share my grant applications and summary statements--successful and unsuccessful--with anyone who asks. As far as scoopage worries, I figure that anything proposed that is not just dotting Is and crossing Ts on presented preliminary data is not particularly likely to ever even happen in my lab (because we will stumble on even more interesting stuff or realize it isn't feasible). And the stuff that is just dotting Is and crossing Ts on presented preliminary data, we are so far ahead of anyone else that who cares if they see it?

This is sort of off-topic, but in case there was any doubt about the reviewer bias against NI/ESI, I have two firsthand anectdata (that keep me up at night in frustration). First anecdote: Two applications, sent to the same SS during the same review cycle. One with noob named PI who spent months meticulously writing every detail, covering every review criteria published in the instructions (including why the Environment is good for NI/ESI -- I think I will ignore this from now on). Second with name established PI, copy/pasting descriptions of scienz written by the same noob (plus other minions); ignoring some criteria, weird formatting, no whitespace. Named noob PI grant is ND and established PI with experiments cobbled together from noob's past work gets a score. Second anecdote: Noob ghost-writes entire grant for established PI (named as co-I), gets funded on A0. Noob writes followup grant as named PI and it's ND. Mostly ad hominem critiques about being inexperienced, random critiques about the writing, "it wanders," while the same goddamned voice/style of the ghost-written grant was praised as "well-written." (I personally think that writing style should be off-limits, this is to fund research, not a goddammed writing workshop at the local community college).

I am considering asking to read redacted summary statements for all NI/ESI and the corresponding (redacted) "Environment" sections from NI/ESI applications and see if one gets dinged by just bringing it to their attention. It's in the instructions, but if the major critiques are ad hominens about being inexperienced (or obvious post-hoc justifications of a bias), why bother bringing your noob status to their attention?

The 2007 NI review policy was: "Support new investigators on R01 equivalent awards at success rates equivalent to that of established investigators submitting new R01 equivalent applications." The current review policy is: "NIH will continue to support new investigators on Type 1 (new), R01 equivalent awards at success rates comparable to that of established investigators submitting Type 1 applications."

As a noob, I'm not there to know wth happens in SS's. I don't know if that policy makes a difference in implementation. But god-damme I wish I knew wth goes on in these reviewers' heads.

At the last MRU at which I worked, we maintained a database of successful applications that faculty members could search by mechanism, IC or key words, then request a PDF containing the application (minus the admin/budget pages) combined with the relevant Summary Statement. The PDFs were locked (no editing/copying, low res printing only) and, while anyone could search the database and request files, they PDFs themselves were made available to faculty only.
As administrator of the database, one of my jobs was to keep the database "fresh" by requesting new applications from faculty. The results of my emailed requests were no response/refusal/agreement in approximately equal proportions. Where people expressed concerns about plagiarism, my response was (i) a plagiarized application is likely to pass before at least some of the same reviewers, based on identical subject matter, so someone would smell a rat, (ii) AFAIK, electronic submission are scanned for excessive similarity to previous submissions, to prevent unsuccessful A1s being submitted as new applications, so an extensively plagiarized application would raise red flags, and (iii), even if by some outlandish chance a plagiarized application, ostensibly written by some git without the wits and track record to actually write an original application, passed these hurdles and was actually funded, the original writer is already funded and is at least a year ahead. I'd be far more concerned by paper plagiarism than grant application plagiarism.
Another aspect is that, AFAIK, since grant applications are written by University employees as part of their work, and submitted by the institution, these fall under "work product" and really belong to the institution, not the investigator. Not that we ever considered pulling out that particular club when asking faculty to hand over their successful applications. Summary Statements are a different case, of course.